Shutter Island
A U.S. Marshal is assigned to investigate the disappearance of a patient at a psychiatric hospital located on a remote island.
This movie will screw with your head. It's also nearly impossible to describe without using spoilers because most plot elements have meanings beyond the obvious; there's a reason most of the TVTropes page is blanked out if you have 'Show Spoilers' turned off. Watching it a second time doesn't really help you untangle things. Watching through, you know something is wrong beyond what is overtly being stated, but you don't know exactly what and in many ways the answers aren't really clear. If you like psychological horror, you'll definitely want to check this one out.
With the lights on.